A rough straight ramp is fixed to horizontal ground. The ramp has length $$\(15 \mathrm{~m}\)$$ and is inclined at an angle $$\(\alpha\)$$ to the ground, where $$\(\tan \alpha=\frac{5}{12}\)$$. The line $$\(A B\)$$ is a line of greatest slope of the ramp, where $$\(A\)$$ is at the bottom of the ramp, and $$\(B\)$$ is at the top of the ramp, as shown in Figure 3. A particle $$\(P\)$$ of mass $$\(6 \mathrm{~kg}\)$$ is projected up the ramp with speed $$\(14 \mathrm{~m} \mathrm{~s}^{-1}\)$$ from $$\(A\)$$ in a straight line towards $$\(B\)$$. The coefficient of friction between $$\(P\)$$ and the ramp is 0.25 (a) Find the work done against friction as $$\(P\)$$ moves from $$\(A\)$$ to $$\(B\)$$. (3) At the instant $$\(P\)$$ reaches $$\(B\)$$, the speed of $$\(P\)$$ is $$\(v \mathrm{~ms}^{-1}\)$$. After leaving the ramp at $$\(B\)$$, the particle $$\(P\)$$ moves freely under gravity until it hits the horizontal ground at the point $$\(C\)$$. Immediately before hitting the ground at $$\(C\)$$, the speed of $$\(P\)$$ is $$\(w \mathrm{~m} \mathrm{~s}^{-1}\)$$ (b) Use the work-energy principle to find (i) the value of $$\(v\)$$, (ii) the value of $$\(w\)$$. (7)

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